initial ordinal वाक्य

उदाहरण वाक्य

1. The least ordinal associated with a given cardinal is called the " initial ordinal " of that cardinal.
2. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial but need not be regular.
3. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not.
4. The axiom of choice is equivalent to the statement that every set can be well-ordered, i . e . that every cardinal has an initial ordinal.
5. In theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ the Von Neumann cardinal assignment as the cardinal's representation.
6. If that initial ordinal is \ omega _ { \ lambda } \,, then the cardinal number is of the form \ aleph _ \ lambda for the same ordinal subscript & lambda;.
7. Then we can apply the axiom of replacement to replace each element of that powerset of " x " by the initial ordinal number of the same cardinality ( or zero, if there is no such ordinal ).
8. In this case, the ordinals 0, 1, \ omega, \ omega _ 1, and \ omega _ 2 are regular, whereas 2, 3, \ omega _ \ omega, and ? ?? are initial ordinals which are not regular.
9. In this case, the ordinals 0, 1, \ omega, \ omega _ 1, and \ omega _ 2 are regular, whereas 2, 3, \ omega _ \ omega, and ? ?? are initial ordinals that are not regular.
10. ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality ( i . e . is equinumerous with its initial ordinal ), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.